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An invariant subspace of a <linear map> $A:V\to V$ is a <vector subspace> $W\subseteq V$ satisfying $A(W)\subseteq W$. For a <group representation>, invariance means $\rho(g)W\subseteq W$ for every represented group element $g$.

Invariance of a <vector subspace> does not mean that each individual <vector> stays fixed. The zero-sum plane in the three-coordinate example is invariant even though the cyclic permutation rotates its nonzero <vectors>. By contrast, the constant-coordinate line is pointwise fixed. Distinguishing these statements prevents a symmetry-preserved mode space from being mistaken for a static mode.


 Invariant subspace (source code)